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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 30th 2012

Stephan Spahn added stuff to plus construction on presheaves

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 30th 2012

Thanks, Stephan!

To address the question of compact objects in Sh ∞(SmoothMfd) there should be an (∞,1)-plus construction, too. Is in this case where the (∞,1)-site is just a 1-site somehow clear how this works?

For $n$-truncated objects it is in principle clear: one has to apply the plus-construction (n+2)-times in a row!

See for instance section 6.5.3 of HTT.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 30th 2012

I have added a brief remark on this in the Idea section. But am out of time now.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeApr 30th 2012
• (edited Apr 30th 2012)

I have added the reference to the famous Heller-Rowe article where the plus construction in the case of abelian sheaves is studied. In the abelian context one usually says Heller-Rowe functor. Interestingly, the theory of Q-categories of Rosenberg has been written out in 1988 to exhibit the common and nontrivial generalization of the functor ${H}_{ℱ}$ in the theory of Gabriel localization and Heller/Rowe functor, as the instances of the same construction. Gabriel localization is also ${H}_{ℱ}^{2}$, just like sheafification. Here $ℱ$ is a Gabriel filter.

• CommentRowNumber5.
• CommentAuthorZhen Lin
• CommentTimeApr 30th 2012

I always thought that the fact that we needed to do $\left(-{\right)}^{+}$ twice had something to do with the fact that the equaliser diagram has two stages, but I never did find a good technical explanation of this point. What is clear, though, is that doing it once is a generalisation of computing ${\stackrel{ˇ}{H}}^{0}$. I imagine these two facts are related via $n$-categories…